Work, Energy and Power

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Work — Force Acting Through Distance:

Work is done when a force causes displacement in the direction of the force. W = F × d × cosθ, where F is the force, d is the displacement, and θ is the angle between the force and the direction of displacement. SI unit is the Joule (J). When θ = 0° (force and displacement in the same direction), W = Fd (maximum). When θ = 90° (force perpendicular to displacement), W = 0. When θ > 90° (force has a component opposing displacement), W is negative.

Examples from everyday life: lifting a bucket vertically upward at constant speed — work done against gravity = mgh. Dragging a sledge along level ground — work done against friction = μmg × d.

Kinetic Energy — Energy of Motion:

KE = ½mv². This is always positive. If you double the mass, kinetic energy doubles. If you double the speed, kinetic energy quadruples (because v² becomes 4 times larger). This is why car accidents at high speed cause disproportionately more damage. SI unit: Joule.

Potential Energy — Energy of Position:

Gravitational PE = mgh (weight × height above a reference level). This depends on where you set zero — any convenient level can be chosen. The change in gravitational PE between two heights depends only on the height difference, not the path taken. Spring PE = ½kx², where k is the spring constant (N/m) and x is the displacement from natural length.

⚡ JAMB Tip: In JAMB, the most common error is using the wrong angle in W = Fd cosθ. Only use cosθ when F and d are explicitly given separately. If the force is parallel to displacement, cosθ = 1. If force is at 30° to the horizontal and you drag an object along the horizontal, the work done = F × d × cos30°.

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🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Power — Rate of Doing Work:

Power = Work done / Time taken = Energy transformed / Time. SI unit: Watt (W). 1 Watt = 1 Joule per second. Other units: 1 kilowatt (kW) = 1000 W, 1 horsepower (hp) ≈ 746 W (used for motors and engines).

Instantaneous power = Force × Instantaneous velocity = Fv. For a car climbing a hill at constant speed, power = total resistive force (weight component along slope + friction + air resistance) × velocity. The maximum speed on a level road is reached when engine power equals the total dissipative forces.

Conservation of Mechanical Energy:

In the absence of non-conservative forces (friction, air resistance), total mechanical energy (KE + PE) is conserved. For a pendulum: at the bottom (maximum speed, minimum height), KE is maximum, PE is minimum. At the extremes ( momentarily at rest, maximum height), KE = 0, PE is maximum. Between these points, energy transforms continuously between kinetic and potential forms.

For a roller coaster car descending a height h from rest: v = √(2gh) at the bottom, regardless of the slope angle or path. This is a powerful shortcut — it avoids calculating acceleration along the track.

Efficiency:

Efficiency (%) = (Useful output energy / Input energy) × 100%. No real machine is 100% efficient because some energy is always lost to friction, heat, or sound. For a machine doing work against friction: Efficiency = (Work output / Work input) × 100% = (Work output / (Work output + Energy lost to friction)) × 100%.

⚡ JAMB Tip: The work-energy theorem states: Net work done = Change in kinetic energy = ½mv² - ½mu². This is particularly useful when forces act over known distances but time is unknown. For a block sliding down a rough inclined plane: net work = mgsinθ × d - μmgcosθ × d = change in KE = ½mv² - 0 (starting from rest).

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Collisions — Elastic and Inelastic:

In a perfectly elastic head-on collision between mass m₁ (velocity u₁) and mass m₂ (velocity u₂ = 0):

• v₁ = ((m₁ - m₂)/(m₁ + m₂)) × u₁
• v₂ = (2m₁/(m₁ + m₂)) × u₁

These equations satisfy both conservation of momentum AND conservation of kinetic energy.

In a perfectly inelastic collision (objects stick together and move as one):

• Combined velocity v = (m₁u₁ + m₂u₂)/(m₁ + m₂)
• Kinetic energy is NOT conserved — some KE is lost to deformation, heat, sound

The coefficient of restitution e = (velocity of separation after collision)/(velocity of approach before collision). For perfectly elastic: e = 1. For perfectly inelastic: e = 0.

Energy Dissipation in Real Systems:

When friction acts over a distance, the work done by friction = μmg × d (for horizontal surface) is converted to heat (not lost — heat is energy, just not mechanical). This is why rubbing your hands together on a cold morning warms them up. The mechanical work done against friction appears as thermal energy.

For a block sliding down a rough incline from height h: v_bottom = √(2gh - 2μmgcosθ × d/m), where d is the distance along the incline (d = h/sinθ). Simplifying: v² = 2gh - 2μgh cotθ. The term 2μgh cotθ represents energy lost to friction.

⚡ JAMB Pattern: JAMB questions on work, energy, and power typically test three things: (1) identifying the correct angle in W = Fd cosθ (a common trick: give you the force at 60° to horizontal and ask for work done while the object moves horizontally — the answer uses cos60°, not cos0°), (2) using energy conservation to find speed at different heights, and (3) calculating power from work/time or F × v. A typical JAMB question: “A body of mass 5 kg falls from a height of 20 m. What is its kinetic energy just before hitting the ground?” Answer: KE = mgh = 5 × 10 × 20 = 1000 J. The velocity would be v = √(2gh) = √(400) = 20 m/s.